## Abstract

The vibration and sound radiation characteristics of a double leaf panel wall with corrugated core are analyzed. The core comprises of repeating unit cells along the length of the panel. Each cell is a combination of a straight beam with uniform cross section and curved beams with varying cross section. The vibroacoustic property of the corrugated panel is analyzed assuming plain strain condition. The analysis presented in the work shows that the proposed sandwich panel design provides broadband sound transmission loss (STL) characteristics. The dispersion analysis, forced response characteristics, and sound radiation characteristics of the structure are presented. It is found that bending frequency band gap obtained from the dispersion analysis is a prominent criterion to achieve a higher STL. The structural and acoustic behaviors of the sandwich panel are analyzed through the spectral finite element model. The STL for the core element for various taper ratios is presented. It is found that a higher taper ratio shifts the STL characteristics to the low frequency range.

## 1 Introduction

Cellular solids have shown great potential in making lightweight thin structures [1], which have application and demand in aerospace, automobile, and real estate. They are being used as a core material in a number of applications such as vibration control [2,3], sound absorption [4], and heat dissipation attribute [5]. In most commercial applications of lightweight panel walls, the core material comprises of foam and cellular solids [6]. The microstructures of foams are randomly oriented and its stochastic nature is responsible for its sound absorbing nature. The interaction of the air and the porous structure of foam is the primary cause of high damping at higher excitation frequencies. The core with a deterministic structure is studied in the past for obtaining its sound transmission loss (STL) [7]. Analytical and theoretical models were developed to predict the sound transmission loss in an isotropic and orthotropic cores [8,9]. In this respect, a periodic truss core panel has been studied by El-Raheb and Wagner [10]. The effect of fluid loading and damping on the sound panel transmission characteristics is studied for varying panel aspect ratio [11]. Ruzzene [7] extended the work with honeycomb core and effects of core geometry on the structural response and acoustic radiation are communicated. To minimize the transmitted sound power, topology optimization techniques were used to design the core [12,13].

The study on the sandwich plate has mostly remained confined to core elements comprising of straight beams having constant area of cross section. The sound transmission characteristics of sandwich double leaf panel having core, comprised of curved beam element with tapered cross section is not reported in the literature so far. In this paper, we present a unified approach that takes advantage of the free wave propagation phenomena of unit cell and its effect on sound transmission characteristics in the double leaf sound panel with tapered curved core (DLPTC). The tapered curved core and the term “corrugated” are used interchangeably. First, we develop the spectral element of the curved and tapered elements. The DLPTC is designed based on the unit cell comprising of the core and the face plate. The free wave vibration analysis of the infinite structure is performed using transfer matrix in conjunction with Floquet–Bloch theorem considering a single-unit cell. The dynamics attribute of the finite DLPTC is then computed using a spectral finite element model. The transmitted sound in the fluid domain is then evaluated using the Rayleigh integral with displacement field as the input. In this work, the fluid domain is assumed to be much less dense than the structure, such as air. As the structure is coupled to a light fluid, it is considered as weak coupling [14]. Thus, the vibration response of the structure is obtained neglecting the fluid coupling. Also, the acoustic response is obtained by using the vibration of structure as boundary condition.

## 2 Theoretical Formulation

The schematic diagram of the DLPTC is shown in Fig. 1. It consists of two face plates and a tapered curved core. The DLPTC is assumed to be infinite in z-direction. There is no variation of strain in the z-direction. As a result, the three-dimensional problem can be treated as a plain strain problem and the DLPTC is represented with a geometry, shown in Fig. 1(b).

Fig. 1
Fig. 1
Close modal

In line with the conventional method of modeling the sandwich panel with cellular cores [7,10,12], the face plate and the core are considered as frame elements. The members of the core are modeled as curved and tapered beams. The taper ratio is defined as the ratio of the thickness of the beams at two ends is h1/h2. The frame element has slenderness ratio given by h2/L = 0.2 and bent angle, ϕ = L/R, where L is the length of the curved beam and R is the radius of curvature. Thus, a specific value of θ, taper ratio h1/h2, and slenderness ratio defines a particular configuration of the square DLPTC, as shown in Fig. 1(b). The top face plate faces the unbounded fluid domain; however, the bottom face plate is subjected to a normally incident uniform harmonically varying pressure wave.

## 3 Formulation of the Dynamic Stiffness Matrix

### 3.1 Dynamic Stiffness Matrix of Curved and Tapered Beams.

The unit cell of the proposed structure is shown in Fig. 1(a). The detailed schematic of the two curved beams used in the unit cell is shown in Fig. 1(d). The curved beams have a uniform taper along its length. The taper is same for both the curved beams. There are two types of curved beams in the unit cell: (1) beam 1, with increasing taper and (2) beam 2, with decreasing taper. c1 and c2 are conicity associated with beams 1 and 2. They are defined as c1 = (h2h1)/h1, c2 = (h2h1)/h2, where h1 and h2 represent the thickness of the beam 1 at s = 0 and s = L, respectively. Using the conicity, the cross-sectional area and moment of inertia of beams 1 and 2 are given by
$A(sj)=A0j(1+(−1)j+1cisjL)nI(sj)=I0j(1+(−1)j+1cisjL)n+2$
(1)
where A0j and I0j are the area and moment of inertia of the cross section at the arc length coordinate sj = 0, where j = 1, 2. Without the loss of generality the force-displacement relation in the beams 1 and 2 are given as [15]
$M=EI(s)(1Rduds−d2wds2)F=EA(s)(wR+duds)V=EI(s)(1Rd2uds2−d3wds3)$
(2)
Similarly, the equilibrium equations are given as
$dVds−FR=−ρA(s)ω2wdFds+VR=−ρA(s)ω2u$
(3)
where, M is the moment of inertia, F is the normal force, V is the shear force across the cross section at a particular arc length s. E is the Young’s modulus, ρ is the density, w and u represent the transverse and longitudinal displacements of the centroidal points. ψ = (u/R) − dw/ds represents the rotation of the cross section.
In this work, the state-space approach to derive the dynamic stiffness matrix is used. Toward this end, we define a state vector for the two curved beams as $y(s)={u,w,ψ,F,V,M}T$. Note, the explicit dependence on s of the right-hand side of Eq. (3) is omitted for brevity. Following the procedure discussed by Lee [16], the state-space equation beam 1 can be written as follows:
$dyds=B1y$
(4)
where, B1 is given by
$B1=[0−1/R01/(EA01g1n)001/R0−1000000001/(EI01g1n+2)−ρA01ω2g1n000−1/R00−ρA01ω2g1n01/R00000010]$
where, g1 = g(s) = (1 + c1s/L).

Note, the matrix B1 is a function of the arc length coordinate s. As such, Eq. (4) cannot be directly integrated to derive the transfer matrix. Thus, to derive the transfer matrix (C1) for beam 1, we implement the Floquet theory [17].

Similarly, the state-space equation for beam 2 can be written as follows:
$dyds=B2y$
where, B2 is given by
$B2=[0−1/R01/(EA02g2n)001/R0−1000000001/(EI02g2n+2)−ρA02ω2g2n000−1/R00−ρA02ω2g2n01/R00000010]$
where, g2 = g2(s) = (1 − c2s/L). Similar to beam 1, we derive the transfer matrix (C2) for beam 2 using the procedure detailed in Ref. [17].

Note, the transfer matrices for beams 1 and 2 are derived in their respective coordinate systems. The transfer matrix of each beam in the unit cell is valid in the local coordinate system of the respective beams. Therefore, we first transform the transfer matrices from local coordinate system to global coordinate system before applying the continuity conditions. A detailed discussion about the transformation can be found in Refs. [3,18]. Using the procedure discussed in Ref. [16], we obtain the global dynamic stiffness matrix from the global transfer matrix. Let $D2,D3,andD4$ be the dynamic stiffness matrices of the curved beams between nodes (4, 5), (2, 5), and (5, 6), respectively (refer to Fig. 1(c)).

### 3.2 Dynamic Stiffness of the Face Plate.

The face plate is modeled using the frame element. The dynamic stiffness matrix of a frame element is derived using spectral element formulation. The governing equation [19] of the flexural and axial wave propagation in the local coordinate system is given as
$EI∂4W(η,t)∂η4+ρA∂2W(η,t)∂t2=0$
(5)
$EA∂2U(η,t)∂η−ρA∂2U(η,t)=0)∂t2$
(6)
where, η is the axial coordinate, U(η, t), W(η, t) is the axial and transverse displacements in the local coordinate system. I and A are the moment of inertia and area of cross section of the straight beam. Considering a harmonic solution in the form U = u(η)eiωt and W = w(η)eiωt, where ω is the free wave frequency, the axial displacement u(η), and transverse displacement w(η) is written in spectral form as
$u=B1sin(k1η)+B2cos(k1η)w=B3sin(k2η)+B4cos(k2η)+B5sinh(k2η)+B6cosh(k2η)$
(7)
where $k1=ωρA/EA$ and $k2=ω(ρA/EI)1/4$ are the axial and flexural wave numbers in the frame element. The expression for axial force F, shear force V(η), and bending moment M(η) in the frame is given as
$F=EAdudη;V(η)=−EId3wdη3;M(η)=EId2wdη2$
(8)
The nodal displacements and slopes of the finite beam element are related to the displacement fields as
${d}={u1w1ψ1u2w2ψ2}={u(0)w(0)w′(0)u(l1)w(l1)w′(l1)}$
(9)
Using Eq. (7), the wave amplitude coefficient vector B = {B1, B2, B3, B4, B5, B6} is given as
${B}=[H1(ω)−1]{d}$
(10)
The bending moment, shear force at nodes 1 and 2 is given by
${Fe}={F1V1M1F2V2M2}={−F(0)−V(0)−M(0)F(l1)V(l1)M(l1)}$
(11)
Using Eqs. (7) and (8) into the above equation we get
${Fe}=[H2(ω)]{B}$
(12)
The value of amplitude wave vector {B} from Eq. (10) is used in Eq. (12) to obtain
${Fe}=[H2(ω)][H1(ω)]−1{d}$
(13)
The dynamic stiffness matrix for the frame element is given as
$De(ω)=[H2(ω)][H1(ω)]−1$
(14)
where, De is the elemental dynamic stiffness in the local coordinate system. Let D1 and D5 be the dynamic stiffness matrices (in the global coordinate system) of the straight beams between nodes (1, 4) and (6, 3), respectively (refer to Fig. 1(c)).

## 4 Matrix Equation for Axial and Flexural Displacements

The continuous displacement field of axial displacement and flexure displacement is obtained using Eq. (13) in Eqs. (5) and (6) as
${uw}=[sin(k1η)cos(k1η)000000sin(k2η)cos(k2η)sinh(k2η)cosh(k2η)][B]=[sin(k1η)cos(k1η)000000sin(k2η)cos(k2η)sinh(k2η)cosh(k2η)][H1(ω)−1]{d}$
The above equation can be rewritten as
${uw}=[N1(η,ω)N2(η,ω)000000N3(η,ω)N4(η,ω)N5(η,ω)N6(η,ω)]{d}$
(15)
where Ni(η, ω) for i=1,6, are the shape functions. The shape function matrix, which is a continuous function of the axial coordinate (η), along with the nodal displacement vector can be used to obtain the displacement of a point anywhere in the frame element.
The bottom layer is subjected to a harmonically varying incident sound pressure loading Pi(η, ω) = Pieiωt, where
$Pi=Peikxx+ikyy=Peik(xsinθ+ycosθ)$
(16)
where, θ is the angle of incidence of the acoustic pressure when calculated with respect to y-axis, k = ω/cair is the wave number of the acoustic pressure wave, and cair is the speed of the sound in air.

### 4.1 Nodal Load Due to Incident Pressure.

The bottom face panel elements are subjected to distributed pressure load P. The equivalent nodal load over the element is obtained using the principle of virtual work [7]. The work done by the distributed load P(η, ω) over the element is given as
$W=∫0l1δwP(η,ω)bdη$
(17)
where δw is the virtual flexural displacement of the bottom face and b is the width of the beam. The virtual flexural displacement δw is interpolated in terms of shape functions [N(η, ω] = {N3(η, ω), N4(η, ω), N5(η, ω), N6(η, ω)} taking arbitrary constants vector δd as δw = [N(η, ω]δd
$W=∫0l1b[N(η,ω)]P(η,ω)dηδd$
(18)
The forces and moments on the nodal points are given by
${V1M1V2M2}=∫0l1b[N(η,ω)]P(η,ω)dη$
(19)
The governing equation of a single frame element in the local coordinate system is given by the following equation
$De(ω){d}=Fe$
(20)
The core, bottom, and upper layer frame elements are assembled in the global coordinate system using the procedure shown in Cook et al. [20] to form the global dynamic stiffness matrix as
$[D(ω)]{d¯}=F$
(21)
The aforementioned equation is used to calculate the nodal displacements of the structure. The nodal displacements of each element are used in Eq. (13) to obtain the displacement over the whole length of the top surface.

## 5 Radiation Pressure in Acoustic Domain

The transmitted sound pressure Pt(r, θ, ω) in the acoustic fluid domain due to incident acoustic pressure Pi on the bottom face of the panel wall is given by Rayleigh’s integral [21] as
$Pt(r,θ,ω)=∫Ltρaω2H01[kR(x)]v(x)dx$
(22)
where, k = ω/cair is the wave number of the acoustic pressure wave, ρa is the density of the fluid domain, $R(x)=(rcosθ−x)2+(rsinθ)2$ is the radial distance between a point on the beam element having velocity v(x) = iωw and the observation point, $H01$ is the Hankel function of the first kind, Lt is the total length of the structure. The transmitted sound power over the receiver surface S* in the far field shown in Fig. 1(b) is given by
$Λt=12ρacairRe(∫0πPt(r,θ,ω)2rdθ)$
(23)
The STL is thus defined as
$STL=10log10ΛtΛi$
(24)
where $Λi=Pi2/2ρacair$ is the incident sound power of the plane pressure wave.

## 6 Numerical Verification

In this section, the numerical verification of the aforementioned methodology is presented. In order to verify the procedure, the transmission loss calculated by Griese et al. for the hexagonal sandwich beam is reproduced using the aforementioned methodology. It is to be noted the hexagonal core elements by Griese et al. [4] were comprised of straight frame elements with no curvature and taper. In the aforementioned methodology, straight beam for the core is obtained by substituting bent angle ϕ = 0 and conicity c1 = 0. The obtained transmission loss is shown in Fig. 2, also overlaid is the transmission loss calculated using the effective medium theory [21]. It is seen the results obtained are in good correlation with both the mass law, spectral element method and those obtained by Griese et al. [4].

Fig. 2
Fig. 2
Close modal

## 7 Dispersion Curve of the DLPTC Unit Cell

Consider the unit cell shown in Fig. 1(c). The dynamic stiffness matrix obtained from an individual curved and tapered element D1, D2, D3, D4, D5 is assembled to obtain the global dynamic stiffness matrix [20]. Thus we get
$[D11D12D13D14D15D16D21D22D23D24D25D26D31D32D33D34D35D36D41D42D43D44D45D46D51D52D53D54D55D56D61D62D63D64D65D66]{p1p2p3p4p5p6}={q1q2q3q4q5q6}$
(25)
where pi = ui, vi, θi and ui, vi, θi refer to axial displacement, transverse displacement, and slope at the ith nodal points. Similarly, the qi = Fi, Vi, Mi represents the axial force, shear force, and bending moments at the ith nodal points for i=1 to 6. The above equation can be rewritten in terms of its left nodes and right nodes as
$[D¯LLD¯LRD¯RLD¯LL]{PLPR}={QLQR}$
(26)
where PL = p1, p2, p3, PR = p4, p5, p6, QL = q1, q2, q3, and QR = q4, q5, q6. The above equation can be rearranged to give transfer matrix relating generalized displacements and forces at the left end to the right end as
${PRQR}=[−D¯LR−1D¯LLD¯LR−1D¯RL−D¯RRD¯LR−1D¯LL−D¯RRD¯LR−1]{PLQL}=[T]{PLQL}$
(27)
where [T] is the transfer matrix. Using the Floquet–Bloch theorem on the state vectors at the left end and right end as
${PRQR}=eiμ{PLQL}$
(28)
Equation (5) reduces into an eigenvalue problem given as
$([T(ω)]−λI){PLQL}=0$
(29)
where, λ = e. In the above equation, the transfer matrix is an 18 × 18 matrix. The solution of the above equation at a particular frequency gives 18 eigenvalues and 18 eigenvectors. Each eigenvectors correspond to the free characteristic wave and the eigenvalues give its corresponding phase constant μ. The phase constant μ is found by taking −i ln(λ). The phase constant μ is complex in nature whose real part of $R(μ)$ gives the propagation constant and the imaginary part $I(μ)$ gives the attenuation constant for each of the 18 characteristic waves. A non-zero attenuation constant $I(μ)$ for a characteristic wave implies that wave undergoes spatial attenuation. The dispersion curve is obtained by plotting the values of propagation constant $R(μ)$ and attenuation constant $I(μ)$ with varying frequency.

## 8 Finite Structure Response and Sound Transmission Loss

### 8.1 Geometry and Material Properties.

The double leaf panel with corrugated core considered is 2 m long and 5 cm thick. The face plate, as well as the core are made up of aluminum (E = 70 × 109 N/m2, ρ = 2700 kg/m3). The maximum thickness of any member of the core is h2 = 2.5 mm while top and bottom face plates is h2/2 = 1.25 mm thick and L = 0.0125 mm as shown in Figs. 1(c) and 1(d). The bent angle of all the core members is taken as ϕ = 30 deg and taper ratio h1/h2 is taken as variable. The out of plane thickness b for all the members is considered to be unity. The fluid medium on both sides is considered to be air having density of air ρa = 1.2 kg/m3 and speed of sound in air cair = 343 m/s. In all the configurations, the panel is placed in a rigid baffle with simply supporting boundary condition.

### 8.2 Panel Response.

The bottom face of the panel is loaded with a normally incident pressure of amplitude equal to one, as shown in Fig. 1(b). First, we consider the configuration in which the bent angle ϕ = 0 and conicity c1 = 0, this implies that all the members of the core element are straight and have a uniform thickness of h2 = 2.5 mm. In Fig. 3(b), the transverse displacement response of the top face plate is presented in terms of root-mean-square velocity vrms. The vrms is defined as
$vrms=[1L∫0L(ωw(x))2dx]1/2$
(30)
The vrms is presented in dB using a reference velocity of 5 × 10−8 m/s [7,10]. Alongside of it in Fig. 3(a), is the dispersion curve obtained using the method presented in Sec. 7. For the frequency interval 4140–7545 Hz, the bending wave mode does not propagate. The frequency range is shown in Fig. 3(a) with gray shade. The mode shape of a characteristic bending wave is calculated using the eigenvector of Eq. (29) and using Eq. (15) to get the intermediate displacement values. The $I(μ)$ is positive for the bending mode, as a result of that these particular wave mode attenuates spatially as it propagates. It is important to note that the structure does not possess complete band gap, as the other characteristic waves propagate in the structure. However, the band exists in different characteristic modes, such as bending, shear, longitudinal, and rotational modes [22].
Fig. 3
Fig. 3
Close modal

In Fig. 3(c), the sound transmission loss through the structure for a normally incident sound pressure field is shown. At low frequency region, there is no significant transmission loss as the bending wave mode propagates, whereas the STL in the bending band gap mode, is predominantly high. The sound transmission loss drops at natural frequency of the simply supported structure, however, the average sound transmission loss over the band gap region is on a higher side when compared to the frequency region where the bending wave mode propagates. In particular, an average 76 dB of STL is observed in the band gap corresponding to the bending mode as shown in Fig. 3(c).

The mode shape of the finite structure response at 500 Hz where the bending wave propagates is shown in Fig. 4(a) and the sound pressure level (SPL) is calculated as
$SPL=20log10(Pt(x,y)20×10−6)$
(31)
The SPL level in semi-infinite fluid domain is shown in Figs. 4(c) and 4(d) for 500 and 4801 Hz, respectively. The structure is placed in the baffle from $x=−1to1m$. It is seen that the sound pressure level in the far field for frequency 500 Hz where the bending mode propagates is large in the far field. For a chosen frequency of 4801 Hz in which the bending mode does not propagate the mode shape is shown in Fig. 4(b), the displacement amplitude is considerably reduced over the top layer and the SPL level at this frequency is very low. Therefore, from a design point of view, the panel needs to be designed so that the band gap in bending mode is maximized.
Fig. 4
Fig. 4
Close modal

### 8.3 Effect of Varying Taper on Sound Transmission Loss.

The parameter taper ratio h1/h2 is varied to understand its effect on the band gap in bending mode. The evolution of bending mode band gap as the conicity parameter is varied is shown in Fig. 5. It is seen that with decrease in the taper ratio the bandwidth of the mode decreases and the band starts at lower frequency. The dispersion curve, root-mean-square velocity, and STL for taper ratios h1/h2 = 0.5, 0.1 are shown in Figs. 6 and 7. For a smaller taper ratio h1/h2, the performance of the panel system is improved in the low frequency interval of 2500–4700 Hz. When compared to the prismatic straight core beams where the band value is 4140–7545 Hz, the tapered core with bent angles provides a significant improvement toward achieving higher STL performance at lower frequency. Thus the corrugated core offers a better acoustic performance and an opportunity to tune the frequency to obtain the desired sound transmission loss performance.

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

## 9 Conclusion

This paper studies the vibration and acoustic performance of the double leaf sound panel with tapered curved core. The double leaf corrugated core is analyzed assuming plain strain condition. The proposed structure presents a viable structure for reduction of sound over a wide frequency of ranges. The parameters influencing the bandwidth over which the sound transmission loss is increased are studied. The analysis is performed by changing the core element geometry through taper ratio and bent angle of the individual element. A spectral finite element model is used to find the vibration response of the structure which is then coupled with the Rayleigh integral to find the pressure field in the semi-infinite acoustic domain. In addition to the dispersion curve of the infinite structure is presented to gain additional insight into the phenomena leading to increased transmission loss.

In this study, we elucidate that the sound transmission is high in those frequencies where the bending mode of free characteristics wave propagates. The frequencies in which the bending wave mode do not propagate, the sound transmission is very low. Hence a core can be suitably designed by designing the structure in such a that the sound frequency falls in the range where the bending wave modes do not propagate. Thus, the band gap in bending mode is a favorable condition for increased sound transmission loss. It is also found that for the corrugated core element, reduction of taper ratio to lower values shift the band gap in bending mode to lower frequency. Thus taper ratio along with bent angle of the core can act as tuning parameter to achieve broadband STL characteristics.

## Acknowledgment

Author acknowledges SERB-NPDF Grant No. PDF/2021/000087 and Indo-Canada IC impact Grant No. DST/INT/CAN/P-03/2020 for supporting the research.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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